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Theorem tgptmd 20341
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tgptmd  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )

Proof of Theorem tgptmd
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2467 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 20339 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( invg `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp2bi 1012 1  |-  ( G  e.  TopGrp  ->  G  e. TopMnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5588  (class class class)co 6284   TopOpenctopn 14677   Grpcgrp 15727   invgcminusg 15728    Cn ccn 19519  TopMndctmd 20332   TopGrpctgp 20333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6287  df-tgp 20335
This theorem is referenced by:  tgptps  20342  tgpcn  20346  tgpsubcn  20352  tgpmulg  20355  oppgtgp  20360  tgplacthmeo  20365  subgtgp  20367  clsnsg  20371  tgpt0  20380  prdstgpd  20386  tsmssub  20414  tsmsxp  20420  trgtmd2  20434  nlmtlm  20965  qqhcn  27636
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